Let $f$ and $g$ be the functions given by $f ( x ) = \frac { 1 } { 4 } + \sin ( \pi x )$ and $g ( x ) = 4 ^ { - x }$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$, as shown in the figure above. (a) Find the area of $R$. (b) Find the area of $S$. (c) Find the volume of the solid generated when $S$ is revolved about the horizontal line $y = - 1$.
: \text { integral }
Let $f$ and $g$ be the functions given by $f ( x ) = \frac { 1 } { 4 } + \sin ( \pi x )$ and $g ( x ) = 4 ^ { - x }$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $S$ is revolved about the horizontal line $y = - 1$.